Editing Prime ideal (section)

===Examples===

* A simple example: In the ring <math>R=\Z,</math> the subset of [[parity (mathematics)|even]] numbers is a prime ideal.
* Given an [[integral domain]] <math>R</math>, any [[prime element]] <math>p \in R</math> generates a [[principal ideal domain|principal]] prime ideal <math>(p)</math>. For example, take an irreducible polynomial <math>f(x_1, \ldots, x_n)</math> in a polynomial ring <math>\mathbb{F}[x_1,\ldots,x_n]</math> over some [[field (mathematics)|field]] <math>\mathbb{F}</math>. [[Eisenstein's criterion]] for integral domains (hence [[Unique factorization domain|UFDs]]) can be effective for determining if an element in a [[polynomial ring]] is [[irreducible polynomial|irreducible]].
* If {{mvar|R}} denotes the ring <math>\Complex[X,Y]</math> of [[polynomial]]s in two variables with [[complex number|complex]] [[coefficient]]s, then the ideal generated by the polynomial {{math|''Y''<sup>&thinsp;2</sup> − ''X''<sup>&thinsp;3</sup> − ''X'' − 1}} is a prime ideal (see [[elliptic curve]]).
* In the ring <math>\Z[X]</math> of all polynomials with integer coefficients, the ideal generated by {{math|2}} and {{mvar|X}} is a prime ideal. The ideal consists of all polynomials constructed by taking {{math|2}} times an element of <math>\Z[X]</math> and adding it to {{mvar|X}} times another polynomial in <math>\Z[X]</math> (which converts the constant coefficient in the latter polynomial into a linear coefficient). Therefore, the resultant ideal consists of all those polynomials whose constant coefficient is even.
* In any ring {{mvar|R}}, a '''[[maximal ideal]]''' is an ideal {{mvar|M}} that is [[maximal element|maximal]] in the set of all [[proper ideal]]s of {{mvar|R}}, i.e. {{mvar|M}} is [[subset|contained in]] exactly two ideals of {{mvar|R}}, namely {{mvar|M}} itself and the whole ring {{mvar|R}}.  Every maximal ideal is in fact prime. In a [[principal ideal domain]] every nonzero prime ideal is maximal, but this is not true in general. For the UFD {{nowrap|<math>\Complex[x_1,\ldots,x_n]</math>,}} [[Hilbert's Nullstellensatz]] states that every maximal ideal is of the form <math>(x_1-\alpha_1, \ldots, x_n-\alpha_n).</math>
* If {{mvar|M}} is a [[Manifold#Differentiable manifolds|smooth manifold]], {{mvar|R}} is the ring of smooth [[real number|real]] functions on {{mvar|M}}, and {{mvar|x}} is a point in {{mvar|M}}, then the set of all smooth functions {{mvar|f}} with {{math|''f''&thinsp;(''x'') {{=}} 0}} forms a prime ideal (even a maximal ideal) in {{mvar|R}}.